SPECTRAL THEORY OF HUBERT 401 C 6 Sp(N), it follows from (1 s/| 1 8 ll(£ °/)(AO - (* °/)WII S e. Since that "^/W) " A(/W)II ^ e and that result. e was arbitrary, this gives the required Finally, to prove (iv), note that h bounded, by virtue of (15.11 .8.1 )3 it i ecause the sequence of norms \\fk(N)\\ is sequence (fk(N) ' x) for x belongin Sen°u£h to prove the convergence of the (7.5.5)). With the notation of (1S%11^vto a total subset of E ((12.15.7.1) and proving the assertion for each Jv *>' We may therefore restrict ourselves to multiplication by the class of th*" ut since fk(Nn) may be identified with , the result follows fr ^^on of /* to SPW) in the sPace *r (15.11 .9) For each universally a closed subspace ofE, stable with res™^1* subset M °/sPW» let E(M) under the orthogonal projector p Pect to N and JV*, which is the image of E the restriction of N to E(M) fc c^} .^ ^M^ (15.10.6). Tfe/2 //ze spectrum of tame<l in M (closure in C). For each n, let EM(M) be the im <PM(^»)- lt is immediately seen thaf^°f E" Under the orth°g°nal projector hence (15.11.5) it is enough to prov ^) « the Hilbert sum of the EM(M), there exists a continuous function ^ ^ Proposition for each Nn . If a ^ M for all C 6 SpW.) (4.5.1). It f^^ Sp(AQ such that ^(Q(a - 0<?M(0 = • AT*--, - - - /* J.L lOll^x-, «^\3/\- 3^-rmv^/ to En(M), then alEn(M) - Ni has an j S that> if Nn is the restriction of A^ En(M). averse equal to the restriction ofg(Nn) to Remark (15.11.10) Note that the ^ for each positive measure ^ on c Proves (ii) in (15.11.8) shows that, by the class of lc in L|(K, M) i$ Wlth compact support K, multiplication that Sp(N) = K (converse of (15 ^ 3^ n°rmal continuous operator N such (15.11.11) LetfbeahomeomorphiSn f subset N ofC containing Sp(jy) r/2e ^ °J a closed subset UofC onto a closed operator N' on E whose spectrum * Te exists a unique normal continuous f(N') = N. S c°ntained in M and which is such that If h is the homeomorphisrn Of >^ virtue of (15.11.8(iii)) we must hav0^^ ^ which is the inverse of/>then bY versely the spectrum of A(N) is/z(Sp(^f' * *(/W) = W; and since con- the result follows. ^ >> ^ M, it follows that/(/z(N)) = AT, andty of a. If J is infinite, then a